Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $r = \dfrac{k - 2}{5k^2 - 50k} \times \dfrac{k^2 - 19k + 90}{k - 2} $
Answer: First factor the quadratic. $r = \dfrac{k - 2}{5k^2 - 50k} \times \dfrac{(k - 10)(k - 9)}{k - 2} $ Then factor out any other terms. $r = \dfrac{k - 2}{5k(k - 10)} \times \dfrac{(k - 10)(k - 9)}{k - 2} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac{ (k - 2) \times (k - 10)(k - 9) } { 5k(k - 10) \times (k - 2) } $ $r = \dfrac{ (k - 2)(k - 10)(k - 9)}{ 5k(k - 10)(k - 2)} $ Notice that $(k - 2)$ and $(k - 10)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac{ \cancel{(k - 2)}(k - 10)(k - 9)}{ 5k\cancel{(k - 10)}(k - 2)} $ We are dividing by $k - 10$ , so $k - 10 \neq 0$ Therefore, $k \neq 10$ $r = \dfrac{ \cancel{(k - 2)}\cancel{(k - 10)}(k - 9)}{ 5k\cancel{(k - 10)}\cancel{(k - 2)}} $ We are dividing by $k - 2$ , so $k - 2 \neq 0$ Therefore, $k \neq 2$ $r = \dfrac{k - 9}{5k} ; \space k \neq 10 ; \space k \neq 2 $